3.1 An Interpretable Machine Learning Model Based on Improved NNs

Although NNs have good utility in processing sample information of BCS, there is still some optimization space. In practical applications, due to different network structures, the prediction accuracy of NNs is also different, and even over-fitting problems occur. That is, due to unreasonable initial state selection and excessive training of the network, the weight threshold falls into the local minimum value ^[11]. Therefore, to optimize the selection of IWT for NNs, this study proposes a method based on genetic algorithm (GA) to divide training set and testing set (TrS & TeS). A comprehensive new method for optimizing the IWT of the network and optimizing dataset partitioning is proposed by integrating the optimization methods of dividing the TrS & TeS designed in the research.

Building a NN prediction model has several steps: input and output parameter selection, normalization processing, selection of network structure, and network training. In the selection of input and output parameters, based on the parameters and performance of BCS, the operating conditions and environmental parameters of BCS can be ignored when establishing a prediction model. The input parameters for selecting the model include speed, operating mode, conveying capacity, conveying distance, etc. ^[12,13].

NNs are a parallel processing system that achieves interpretability of parameters through normalization processing. This study selected $[-1$, $1]$ normalization in machine learning to normalize the parameters, and the expression function is Eq. (1).

In Eq. (1), $x_{\max } $ and $x_{\min } $ are the max and min values of the sample. $x$ and $\bar{x}$ represent vectors before and after normalization. The selection of NNs directly affects their predictive performance. In the study of BCS performance, the speed, operation mode, conveying capacity, conveying distance, environmental temperature and other influencing factors of belt conveyors vary. Therefore, before using a belt conveyor, it is necessary to have a detailed understanding of the performance parameters of the equipment in order to ensure its normal operation and safe use.

The three-layer feedforward network structure of NNs includes input, output, and hidden layers, demonstrating good performance. This three-layer network structure is selected as a prediction study for the performance of BCS ^[14]. Based on the machine performance and emission data that can be collected by the belt transportation system equipment, the input parameters of the neural network prediction model are ultimately determined to be: speed, power, rail pressure, and fuel injection timing; The output parameters to be predicted include: fuel consumption rate, maximum burst pressure, burst pressure angle, turbocharger speed, front vortex temperature, and smoke level. Due to the fact that the neurons in hidden layer (HLN) are an essential structural parameter affecting network performance, analysis and research were conducted on distinctive quantity of neurons. The input layer is equipped with $m$ input nodes, and any input signal is expressed as $x_{i} $. The hidden layer has l hidden neurons, and any of output layer neuron (OLN) is denoted by t. The output layer owns $n$ output neurons. The activation transfer function is the key of NNs and determines their utility, as shown in Eq. (2).

In Eq. (2), $f(t)$ is a threshold function that reflects the excitation or inhibition of neurons. The network training adopts an error backpropagation algorithm, sets a set of samples, and calculates the connection weight matrix (CWM) in the layer of input and hidden, as shown in Eq. (3).

The CWM between the output and the hidden layers is calculated as shown in Eq. (4).

The thresholds $\theta ^{1} $ and $\theta ^{2} $ of HLN and OLN are calculated, as shown in Eq. (5).

The output of HLN is obtained from the forward propagation of NN working signals, as shown in Eq. (6).

In Eq. (6), $l$ represents the amount of HLN. The expected error based on the OLN output is obtained as shown in Eq. (7).

In Eq. (7), $\theta _{j}^{1} $ and $\theta _{k}^{2} $ represent the thresholds of the corporate HLN and OLN. Next, the error signal is backpropagated and the link values are modified layer by layer. Finally, the partial derivative of the connecting weight value to the error is obtained, as shown in Eq. (8).

Due to overtraining, the network falls into local minima, resulting in overfitting and affecting generalization ability. Fig. 1 shows the NN process.

The ``timely termination'' is adopted to optimize the process to define target errors to a small extent and provide sufficient training space. This method monitors the timely error changes of the test set. When the network suffers from a local min and the test set error cannot jump out of the local minimum within the specified cycles numbers, the training will be terminated and the optimal state of the set will be output.

Fig. 1. The training process of neural networks.

3.2 Comprehensive Improvement of NNs and SVM for Predicting High Performance BCS

Due to changes in network structure, the predictive performance of NNs varies. Therefore, it is necessary to randomly define the IWT and divide the TrS & TeS according to a certain proportion, and repeat the training of the network multiple times to analyze its performance ^[15]. The performance parameters of commonly used evaluation and prediction models are shown in Eqs. (9) and (10).

The calculation of mean square error (MSE) and mean absolute percentage error (MAPE) is Eq. (10).

In Eqs. (9) to (10), $n$ represents the samples. $\hat{y}_{i} $ and $y_{i} $ represent the target and predicted values of the $i$-th sample. NNs provide direction for performance prediction of BCS. However, NNs are all influenced by certain factors and exhibit certain instability, so it is necessary to optimize them ^[16]. When using the "timely termination" method to solve the overfitting problem of neural networks, it is necessary to divide the training samples into a training set and a validation set. The training set is used for training network weights and thresholds, while the validation set is used to detect changes in training errors, thereby determining the optimal number of training cycles. Therefore, the division of the training set and validation set will also affect the predictive performance of the trained network. Therefore, a genetic algorithm based optimization algorithm for dividing the training and validation sets is proposed in this study. This chapter will compare and study the existing optimization methods for initial weight thresholds of neural networks using genetic algorithms with the neural network training and validation set optimization algorithm proposed in this paper. By integrating the two optimization algorithms, a new comprehensive optimization algorithm for initial weight values of neural networks and optimization algorithms for training and validation sets will be proposed. In traditional GA, a fitness function is constructed based on the objective function of the problem to be optimized, an initial population is created, followed by genetic coding, evaluation, selection, and calculation of the already encoded population, and multiple iterations are performed. After the above steps, the optimal adaptive individual is the optimal solution of the problem ^[17]. Fig. 2 shows the flowchart of the basic GA.

Fig. 2. Genetic algorithm flowchart.

Firstly, to generate an initial population randomly, calculate the fitness of each individual, and then evaluate their ability to adapt to the environment through a defined fitness function. Selecting individuals based on fitness and reproduce the next generation, followed by genetic operations, including crossover and mutation, to ultimately test the termination conditions of the algorithm. Optimizing the division of the TrS & TeS of the NN, using binary encoding to achieve individual encoding. On this basis, this study proposes a sample partitioning method based on chromosome gene information. The specific optimization population initialization method, encoding, and genetic operation steps are as follows. Initializing the population, marking the sorted training samples with an index number, and dividing them into a TrS & TeS proportionally according to random sorting, to obtain a random quantity of individuals in the given population. To assign the index numbers of the obtained individuals and generate chromosomes. 1 represents the training set sample, and 0 represents the test set sample. The new genetic operation involves adjusting the mutation and crossover operators, as shown in Fig. 3.

Fig. 3. Genetic manipulation.

The crossover step first decodes the parent chromosome to obtain the set to which the sample belongs, and then integrates the test set samples of the decoded two individuals. Randomly to select a certain number of samples as new test set samples, while the rest are training set samples. Based on the new TrS & TeS samples, recoding and obtaining offspring chromosomes. The mutation operation involves randomly selecting a certain amount of training and testing samples for equal sample exchange ^[18]. Based on the optimization of the IWT and test set partitioning of NNs mentioned above, this study proposes a comprehensive NN optimization method combining the two, and establishes a high-performance prediction model for BCS. By comparing the initial weight threshold and optimization methods for training and validation sets of neural networks based on genetic algorithms, and considering that both optimization methods can have a certain optimization effect on the performance of the network. Therefore, a comprehensive neural network optimization algorithm is proposed, which combines the optimization objectives of the above two methods. In the main line of the optimization methods for the training and validation sets with obvious optimization effects, the initial weight threshold optimization of each individual's network is added to achieve comprehensive optimization of the neural network, in order to find the best network state to predict the response of the belt transportation system. The comprehensive process of the algorithm is Fig. 4.

Fig. 4. Comprehensive optimization algorithm flowchart.

Firstly, an initial population with different combinations of training and validation sets is generated. Then, before optimizing the training and validation sets, optimization of the initial network weight threshold for each individual is added. After optimizing the initial weight threshold of each individual in the network, the optimal network performance for each training and validation set individual is obtained, and then genetic evolution is performed on them to generate a new population. Because the partition of the training set and validation set is different, the corresponding optimal initial weight threshold of the network is also different. Therefore, before the evolution of each generation of new populations, the network initial weight and threshold will also be optimized. The termination condition for the final evolution is still the maximum evolution generation. Once the maximum evolution generation is reached, a network with the best partition of training and validation sets is output, and the initial weight threshold of the network is also its corresponding optimal value.

Both optimizations in Fig. 4 are completed by GA, and the fitness function is shown in Eq. (11).

In Eq. (11), $\hat{y}_{i} $ is the estimated value of $y_{i} $, $\bar{y}$ is the average value of $y$, $SSE$ is the sum of squared residuals, and $SST$ is the sum of squared total deviations.

After optimization, the system performance was predicted and SVM parameters were selected in combination with FSA. FSA is an intelligent optimization algorithm based on fish behavior, which gradually optimizes by updating its own position during each iteration process ^[19]. The current state of the $i$-th fish is set to $X_{i} $, with a fitness of $Y_{i} $. Each fish selects a state at random, as shown in Eq. (12).

In Eq. (12), $Rand()$ is a random number generated between 0 and 1. $X_{i} $ represents the current state of the $i$-th fish. $\left\| X_{v} -X_{i} \right\| $ represents the distance between $X_{v} $ and $X_{i} $. $X_{i\mid next} $ represents the next state of the $i$-th fish. FSA mimics the way fish flock, allowing them to move towards the center while limiting crowding and avoiding overcrowding, as shown in Eq. (13).

In Eq. (13), with one's own position as the center, the number of fish within the perception range is $N_{j} $, forming a set $S_{j} $. If set $S_{j} $ is an empty set, indicating that there are fish within the perception range of the $i$-th fish, then calculate the center position $X_{center} $ of the set according to Eq. (14) and calculate the fitness value $Y_{center} $ of the center position.

If tail chasing behavior occurs, it indicates that each fish is pursuing the closest fish with the highest fitness ^[20]. If $X_{\min } $ is the center, the number of fish $N_{f} $ within the perception range satisfies Eq. (15).

In Eq. (15), $X_{i} $ searches for the fish $X_{\min } $ with the best fitness within its perception range, and its fitness value is $Y_{\min } $. Then, it is determined that if $Y_{\min } >Y_{i} $, foraging behavior will be performed, and vice versa, Eq. (15) is satisfied. If it indicates that the location has good adaptability and is not too crowded, then execute Eq. (16) to move forward in the direction of fish $X_{\min } $ with the best adaptability. Otherwise, perform foraging behavior.

In Eq. (16), $\left\| X_{v} -X_{i} \right\| $ represents the distance between $X_{v} $ and $X_{i} $. Fig. 5 shows FSA process.

When SVM is used for BCS performance prediction, its prediction performance may vary due to differences in kernel function parameters and penalty factors. And for different prediction targets, the distribution also varies ^[21]. When using FSA to optimize SVM, the algorithm can accurately and efficiently optimize, and the optimization is stable. Regardless of the initial population distribution state, it can quickly converge to the optimal region. The fish species in each iteration are concentrated towards the optimal region. The algorithm can obtain a more detailed search in the optimal region ^[22]. However, to avoid the algorithm falling into local optima, restrictions have been placed on the crowding degree of the fish school.

Fig. 5. Flowchart of fish school algorithm.

3.3 Overview of Integration of Multiple Technologies

In order to optimize the selection of initial weight thresholds for neural networks, a study was conducted on the partitioning of training and testing sets based on genetic algorithms, combined with the research designed optimization methods for partitioning training and testing sets. Firstly, optimize the initial weight threshold and dataset partitioning of the network. The establishment of a neural network prediction model includes several steps, including the selection of input and output parameters, normalization processing, selection of network structure, and network training. By randomly defining initial weight thresholds and dividing the training and testing sets into a certain proportion, the network is trained multiple times to analyze its performance; By comparing the initial weight thresholds and optimization methods of neural network training and validation sets based on genetic algorithms, it was found that both optimization methods have a certain optimization effect on the performance of the network. Therefore, combining the optimization objectives of the above two methods. In the main line of optimization methods, for the training and validation sets with obvious optimization effects, the initial weight threshold optimization of each individual network was added to achieve comprehensive optimization of the neural network, thereby finding the best network state to predict the response of the belt transportation system.

4.1 Preparation and Design of Experimental Data

A comprehensive study on the optimization of IWT and training set test sets for NNs was conducted. 10 sets of IWT and ten sets of training set and validation set (Ts-Vs) partitioning combinations were randomly generated in the experiment. Using the optimization method of IWT to generate 10 initial populations. For the training set and validation set optimization methods, the IWTs are 10 sets of random IWTs, respectively. Each population has the same combination of 10 Ts-Vs. Based on this, the optimization effects of two optimization methods on NNs were compared. The optimization of NNs selects a network structure with relatively stable performance and four neurons in the hidden layer. The maximum evolutionary algebra set in the experiment is 100 generations, and the population size remains the same as the initial population size at 10. Table 1 shows the relevant parameters of the experiment.

4.2 Analysis of Predictive Performance of Predictive Models

For the evaluation of the initial population, Fig. 6 shows the statistical results of the predictive performance for each initial population. Fig. 6(a) shows the IWT optimization network prediction performance distribution state. Fig. 6(b) shows the distribution status of network prediction performance with a combination of random training set and validation set partitioning. In Fig. 6(a), there is a significant difference in the distribution of the box line structure, indicating that both optimization methods will have a certain impact on the performance of the network. Fig. 6(b) shows the average and minimum values of each initial population, used to analyze the trend of changes between different populations. Compared to the data in Figs. 6(a) and Fig. 6(b), it can be observed that compared to groups with different IWTs, the differences between groups with different combinations of Ts-Vs partitioning are much greater. In Fig. 6(a), there is a significant difference between each box and the overall moving average. The box distribution in Fig. 6(b) is uniform, and there is not much difference between the average and minimum values. The minimum values of all populations are significantly lower than the overall moving average. The minimum values of all populations in the network are significantly lower than the overall moving average.

Fig. 6. Initial population performance evaluation of IWT optimization and training set test set optimization.

Fig. 7 shows the optimization trajectories of different populations under two optimization methods. Fig. 7(a) shows the trajectory of optimizing the IWT of the network under different combinations of Ts-Vs partitioning. With the increase of evolutionary algebra, network performance gradually decreases, indicating the effectiveness of optimization algorithms. Dividing the Ts-Vs is more effective than optimizing the IWT. Due to the differences between the initial group of Ts-Vs, there may be significant deviations in the optimization process of the IWT. Therefore, the optimization process usually takes longer to converge. Fig. 7(b) shows the optimization trajectories of different populations under different IWTs by dividing the Ts-Vs. As shown in Fig. 7(b), optimizing populations with different initial network weight thresholds through Ts-Vs helps to quickly converge to the optimal value and ensure relative stability in the later evolution stage. All evolutionary processes reach their optimal values within 17 generations.

Fig. 7. Optimization trajectories of different populations under two optimization methods.

Fig. 8 shows the synthesis of the final optimization amplitude of two optimization algorithms on network performance, and compares the optimization effects of the two optimization methods on network performance. Fig. 8(a) represents the optimization amplitude of two methods for different initial populations. Fig. 8(b) is the overall average of the two optimization methods relative to the initial population. Fig. 8(a) indicates that both optimization methods have improved the predictive performance of the network to varying degrees. There is a significant difference between the second and eighth red columns in Fig. 8(b). This indicates that the predictive ability of the network after optimizing through weight threshold has not yet reached the average level of all individuals in the initial population. Compared to the average level of all individuals in the initial population, Ts-Vs optimization can more effectively improve the performance of the network than optimizing the IWT.

Fig. 9 shows the evolutionary trajectory of the proposed comprehensive optimization algorithm for 100 generations. The optimization algorithm stably converges to the optimal value through only 14 generations of evolution, resulting in an optimal network prediction MSE of 0.011678. Previously, the average network prediction MSE of all individuals in the initial population was 0.016845. Therefore, using this comprehensive optimization algorithm to optimize the prediction performance of the network reaches 29.6%, and the optimization effect of the NN is superior to both optimization algorithms.

Fig. 8. Comparing the optimization effects of two optimization methods on network performance.

Fig. 9. Evolutionary trajectory of comprehensive optimization algorithm.

Table 2 shows the time occupied by the four calculation methods. The training of the prediction model takes the longest time, and the parameters of it were optimized during the establishment of the model. The optimization process requires repeated training and evaluation of the prediction model, which takes up a long time. However, the total time required for optimizing the performance prediction of BCS is only 5.991 seconds, which is relative to the highest speed of 1750 r/min.

After obtaining a set of optimization results, in order to ensure the accuracy and precision of the optimization results, the simulation model will be used again to verify the accuracy of the optimization results. If the accuracy does not reach the set goal, the new data obtained from the simulation model can be added to the previously obtained database, and all the obtained simulation data can be used to reinforce the support vector machine prediction model again. Then, the support vector machine after reinforcement training is used again to optimize and validate the diesel engine, until the optimization results that meet the accuracy requirements are obtained. In this way, reinforcement learning and optimization of the prediction model can be synchronized, ensuring optimization accuracy while reducing the number of simulation calculations and avoiding waste caused by excessive data. Fig. 10 shows the combination of SVM reinforcement learning and BCS operation parameter optimization process. Fig. 10(a) describes the variation of SVM response prediction accuracy with each BCS reinforcement training. The overall prediction accuracy steadily improves with the increase of reinforcement training. Overall, the proposed reinforcement training method can significantly improve the accuracy of SVM and ensure the accuracy of optimization results. Fig. 10(b) shows the continuously improving performance of BCS simulation throughout the entire optimization process. After the 10th reinforcement training, the total number of simulations was 65, and the prediction error percentage (PEP) of SVM decreased to 0.99%.

Fig. 10. The combination of SVM reinforcement learning and the optimization process of operating parameters in BTS.